[Superconnexion et noyaux de Bergman en famille]
Nous annonçons des résultats sur le développement asymptotique du noyau de Bergman en famille. L'idée principale est d'utiliser le formalisme des superconnexions comme dans la preuve du théorème de l'indice local en famille.
We establish an asymptotic expansion for families of Bergman kernels. The key idea is to use the superconnection formalism as in the local family index theorem.
Accepté le :
Publié le :
Xiaonan Ma 1 ; Weiping Zhang 2
@article{CRMATH_2007__344_1_41_0,
author = {Xiaonan Ma and Weiping Zhang},
title = {Superconnection and family {Bergman} kernels},
journal = {Comptes Rendus. Math\'ematique},
pages = {41--44},
publisher = {Elsevier},
volume = {344},
number = {1},
year = {2007},
doi = {10.1016/j.crma.2006.11.013},
language = {en},
}
Xiaonan Ma; Weiping Zhang. Superconnection and family Bergman kernels. Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 41-44. doi : 10.1016/j.crma.2006.11.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.11.013/
[1] Curvature of vector bundles associated to holomorphic fibrations, 2005 | arXiv
[2] The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math., Volume 83 (1986) no. 1, pp. 91-151
[3] Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants, Comm. Math. Phys., Volume 115 (1988) no. 2, pp. 301-351
[4] Higher analytic torsion forms for direct images and anomaly formulas, J. Algebraic Geom., Volume 1 (1992) no. 4, pp. 647-684
[5] The asymptotics of the Ray–Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys., Volume 125 (1989) no. 2, pp. 355-367
[6] On the asymptotic expansion of Bergman kernel, J. Differential Geom., Volume 72 (2006) no. 1, pp. 1-41 (announced in C. R. Math. Acad. Sci. Paris, 339, 3, 2004, pp. 193-198)
[7] Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999, pp. 13-33
[8] Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math., Volume 24 (1987) no. 2, pp. 227-252
[9] The
[10] Generalized Bergman kernels on symplectic manifolds, C. R. Math. Acad. Sci. Paris, Volume 339 (2004) no. 7, pp. 493-498 (The full version:) | arXiv
[11] Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics, vol. 254, Birkhäuser Boston, Boston, MA, 2007
[12] X. Ma, W. Zhang, Superconnection and family Bergman kernels, in press
[13] The Monge–Ampère operator and geodesics in the space of Kähler potentials, 2005 | arXiv
[14] Complex Monge–Ampère and symplectic manifolds, Amer. J. Math., Volume 114 (1992) no. 3, pp. 495-550
- Toeplitz operators and the full asymptotic torsion forms, Journal of Functional Analysis, Volume 286 (2024) no. 3, p. 110210 | DOI:10.1016/j.jfa.2023.110210
- The asymptotics of the holomorphic analytic torsion forms, Journal of the London Mathematical Society, Volume 108 (2023) no. 1, p. 80 | DOI:10.1112/jlms.12741
- Superconnection and family Bergman kernels, Mathematische Annalen, Volume 386 (2023) no. 3-4, p. 2207 | DOI:10.1007/s00208-022-02438-0
- Geometric quantization of symplectic maps and Witten's asymptotic conjecture, Advances in Mathematics, Volume 387 (2021), p. 107840 | DOI:10.1016/j.aim.2021.107840
- The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle, Geometriae Dedicata, Volume 214 (2021) no. 1, p. 489 | DOI:10.1007/s10711-021-00625-y
- A remark on characterizations of Griffiths positivity through asymptotic conditions, International Journal of Mathematics, Volume 32 (2021) no. 11 | DOI:10.1142/s0129167x21500877
- Second variation of Selberg zeta functions and curvature asymptotics, Annals of Global Analysis and Geometry, Volume 57 (2020) no. 1, p. 23 | DOI:10.1007/s10455-019-09687-4
- Geometric quantization of Hamiltonian flows and the Gutzwiller trace formula, Letters in Mathematical Physics, Volume 110 (2020) no. 7, p. 1585 | DOI:10.1007/s11005-020-01267-z
- ASYMPTOTIC TORSION AND TOEPLITZ OPERATORS, Journal of the Institute of Mathematics of Jussieu, Volume 16 (2017) no. 2, p. 223 | DOI:10.1017/s1474748015000171
- Positivity and vanishing theorems for ample vector bundles, Journal of Algebraic Geometry, Volume 22 (2012) no. 2, p. 303 | DOI:10.1090/s1056-3911-2012-00588-8
- Asymptotic Expansion of the Bergman Kernel, Holomorphic Morse Inequalities and Bergman Kernels, Volume 254 (2007), p. 175 | DOI:10.1007/978-3-7643-8115-8_5
- Kodaira Map, Holomorphic Morse Inequalities and Bergman Kernels, Volume 254 (2007), p. 211 | DOI:10.1007/978-3-7643-8115-8_6
- Positivity of direct image bundles and convexity on the space of Kahler metrics, arXiv (2006) | DOI:10.48550/arxiv.math/0608385 | arXiv:math/0608385
Cité par 13 documents. Sources : Crossref, NASA ADS
Commentaires - Politique